Ex 6.1,15 - Chapter 6 Class 12 Application of Derivatives

Transcript

Ex 6.1,15 The total cost C(𝑥) in Rupees associated with the production of 𝑥 units of an item is given by 𝐶(𝑥) = 0.007𝑥3 – 0.003𝑥2 + 15𝑥 + 4000. Find the marginal cost when 17 units are produced. Since Marginal Cost is Rate of change in total cost w.r.t No of units produced Let M C be the marginal cost & C 𝑥﷯ is the total cost & 𝑥 be the no of units produced So, MC = 𝑑 𝑐 𝑥﷯﷯﷮𝑑𝑥﷯ It is Given that total cost C 𝑥﷯=0.007 𝑥﷮3﷯−0.003 𝑥﷮2﷯+15𝑥+4000 & we need to find Marginal cost when 17 units produced i.e. MC at 𝑥=17 Now MC = 𝑑 𝐶 𝑥﷯﷯﷮𝑥﷯ MC = 𝑑 0.007 𝑥﷮3﷯−0.003 𝑥﷮2﷯+15𝑥+4000﷯﷮𝑑𝑥﷯ MC = 𝑑 0.007 𝑥﷮3﷯﷯﷮𝑑𝑥﷯ − 𝑑 0.003 𝑥﷮2﷯﷯﷮𝑑𝑥﷯+ 𝑑 15𝑥﷯﷮𝑑𝑥﷯+ 𝑑 4000﷯﷮𝑑𝑥﷯ MC = 0.007 𝑑 𝑥﷮3﷯﷯﷮𝑑𝑥﷯ −0.003 𝑑 𝑥﷮2﷯﷯﷮𝑑𝑥﷯ +15 𝑑 𝑥﷯﷮𝑑𝑥﷯+0 MC = 0.007 × 3 𝑥﷮2﷯−0.003 ×2𝑥+15 MC = 0.0021 𝑥﷮2﷯−0.006𝑥+15 We need to find MC when 𝑥=17 Putting 𝑥=17 MC = 0.021 17﷯﷮2﷯−0.006 17﷯+15 MC = 6.069 – 0.102 + 15 MC = 20.967 Hence the Required Marginal cost is Rs. 20.967 when 𝒙=𝟏𝟕